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Let G is a connected graph. A rainbow k-coloring on G is a function c : E(G) → {1, …, k for k ∈ ℕ where for any two vertices u and v in V, there is a path which all edges have no same color. A path that all edges have no the same color is called a rainbow path. Let k is the smallest positive integer that is needed to make G be rainbow connected, then k is called by the rainbow connection number of G symbolized by rc(G). There are many researches on rainbow connection number of operation graph classes that have been done. In this paper, we use join product and strong product as operations on sun graph cn⊙ K̅1 and path graph Pm for determining the rainbow connection number. [ABSTRACT FROM AUTHOR]

Supposed G be a simple graph with a set of vertices (G), set of edges E(G), with | E(G)| = q. A graph G is a harmonious graph if there is an injective function f*: (G) → ℤq, such that the induced function f*: E (G) → ℤq defined by f*(xy) = f(x) + f(y), ∀xy ∈ E(G) is a bijective function. The function f is called harmonious labeling of G. It was known that a complete graph Kn is harmonious only for n ≤ 4 In this paper, we investigate the existence of harmonious labeling of the graphs from the corona operation between a complete graph K4 and Kn k̅n, and also between K5 and k̅n. [ABSTRACT FROM AUTHOR]